24.01.2025

Lorenzo Taggi: Dimensional Phase Transition in Random Walk Loop Soup

We consider a system of closed random walk trajectories interacting by mutual repulsion. This probabilistic model is motivated by its connections to statistical mechanics models, such as the Bose gas, the double dimer model, or the spin O(N) model. We prove the occurrence of macroscopic loops in dimension d>2 (joint with A. Quitmann, 2022) and the absence of macroscopic loops in dimension d=2 (joint with W. Wu, 2024). The talk will primarily focus on the two-dimensional case, which is analyzed using a novel complex spin representation and the derivation of a Mermin-Wagner theorem for complex measures.

Benedikt Jahnel: Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties

In this talk, we consider irreversible translation-invariant interacting particle systems on the d-dimensional hypercubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs with respect to the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure with respect to the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems and joined work with Jonas Köppl.